Data Description¶
- The data contains features extracted from the silhouette of vehicles in different angles. Four "Corgie" model vehicles were used for the experiment: a double decker bus, Cheverolet van, Saab 9000 and an Opel Manta 400 cars.
- This particular combination of vehicles was chosen with the expectation that the bus, van and either one of the cars would be readily distinguishable, but it would be more difficult to distinguish between the cars.
Objective¶
- Find relationship between various attributes (Independent variables) and choose carefully which all attributes have to be a part of the analysis and why
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%%html
<style>
/* Jupyter */
.rendered_html table,
/* Jupyter Lab*/
div[data-mime-type="text-markdown"] table {
margin-left: 0
}
</style>
Import the necessary Libraries¶
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# Basic Libaries
import numpy as np # linear algebra
import pandas as pd # data processing, CSV file I/O (e.g. pd.read_csv)
import warnings
warnings.filterwarnings('ignore')
## Data Visualization Libraries
import seaborn as sns # data plots and visualization
sns.set(style="white")
sns.set(color_codes=True)
import matplotlib.pyplot as plt # data plots
%matplotlib inline
from IPython.display import HTML
Read the data as a data frame¶
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vehicle = pd.read_csv('vehicle.csv')
vehicle.head(10)
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Shape of the data¶
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vehicle.shape
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Data type of each attribute¶
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vehicle.dtypes
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Data Attributes¶
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vehicle.columns
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Vehicle data contains 846 observations with 19 variables, all numerical and one categorical defining the class of the objects.
The following table gives a brief introduction of all features.
Attribute Information¶
| Attribute | Data Type | Description |
|---|---|---|
| compactness | Numeric | Compactness (Average perimeter ** 2/Area) |
| circularity | Numeric | Circularity (Average radius ** 2/Area) |
| distance_circularity | Numeric | Distance Circularity (Area/Average distance from border ** 2) |
| radius_ratio | Numeric | Radius Ration (Maximum radius - Minimum radius/Average radius) |
| pr.axis_aspect_ratio | Numeric | Principal Axis Aspect Ratio (Minor axis/Major axis) |
| max.length_aspect_ratio | Numeric | Maximum Length Aspect Ratio (Length perpendicular maximum length/ Maximum length) |
| scatter_ratio | Numeric | Scatter Ratio (Inertia about minor axis/Inertia about major axis) |
| elongatedness | Numeric | Elongatedness (Area/Shrink width** 2) |
| pr.axis_rectangularity | Numeric | Pricipal Axis Rectangularity (Area/ Principal axis length * Principal axis width) |
| max.length_rectangularity | Numeric | Max Length Rectangularity (Area/Maximum length * Length perpendicular to this) |
| scaled_variance | Numeric | Scaled Variance Along Major Axis (2nd order moment about minor axis/Area) |
| scaled_variance.1 | Numeric | Scaled Variance Along Minor Axis (2nd order moment about major axis/Area) |
| scaled_radius_of_gyration | Numeric | Scaled Radius of Gyration (Maximum variance + Minimum variance/Area) |
| scaled_radius_of_gyration1 | Numeric | Scaled Radius of Gyration 1 (Maximum variance + Minimum variance/Area) |
| skewness_about | Numeric | Skewness |
| skewness_about.1 | Numeric | Skewness About Major Axis (3rd order moment about major axis/sigma_minor** 3) |
| skewness_about.2 | Numeric | Skewness About Minor Axis (3rd order moment about minor axis/sigma_major** 3) |
| hollows_ratio | Numeric | Hollows Ratio (Area of hollows/Area of bounding polygon) where sigma_maj 2 is the variance along the major axis and sigma_min 2 is the variance along the minor axis, and area of hollows = area of bounding poly-area of object |
| class | Categorical | Number of Car Classes (3): Car, Bus, Van |
Exploratory Data Analysis¶
Check for missing value, incorrect data, duplicate data and perform imputation or remove records as necessary¶
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vehicle.info()
As the number of non null observations are not equal to the number of entries, there are likely chance of presence of missing values.
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#Create a new function:
def num_missing(x):
return sum(x.isnull())
#Applying per column:
print("Missing Values in Attributes (if any):")
print(vehicle.apply(num_missing, axis=0))
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# Missing Data correction by dropping null values
vehicle.dropna(inplace=True)
vehicle.shape
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Out of 846 records, 33 records were dropped with NaN values. Now only 813 records remain. \ Note: Dropping is only advised to be used if missing values are few (say 0.01–0.5% of our data). Percent is just a rule of thumb.
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## Cross Validate for presence of any missing values
#Create a new function:
def num_missing(x):
return sum(x.isnull())
#Applying per column:
print("Missing Values in Attributes (if any):")
print(vehicle.apply(num_missing, axis=0))
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# Check if there are rows having ?
vehicle[vehicle['circularity']=="?"]
vehicle[vehicle['distance_circularity']=="?"]
vehicle[vehicle['radius_ratio']=="?"]
vehicle[vehicle['pr.axis_aspect_ratio']=="?"]
vehicle[vehicle['scatter_ratio']=="?"]
vehicle[vehicle['elongatedness']=="?"]
vehicle[vehicle['pr.axis_rectangularity']=="?"]
vehicle[vehicle['scaled_variance']=="?"]
vehicle[vehicle['scaled_variance.1']=="?"]
vehicle[vehicle['scaled_radius_of_gyration']=="?"]
vehicle[vehicle['scaled_radius_of_gyration.1']=="?"]
vehicle[vehicle['skewness_about']=="?"]
vehicle[vehicle['skewness_about.1']=="?"]
vehicle[vehicle['skewness_about.2']=="?"]
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No Question Mark ("?") sign present in any data attribtes
Check for duplicate data
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dups = vehicle.duplicated()
print('Number of duplicate rows = %d' % (dups.sum()))
vehicle[dups]
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No duplicate rows exists and hence no duplicate removal step is required
Data Summary¶
Descriptive Statistics for Categorical Variables¶
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vehicle.describe(include = ["object"]).transpose()
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Descriptive Statistics for Numerical Variables¶
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df = vehicle.describe().transpose()
dfStyler = df.style.set_properties(**{'text-align': 'left'})
dfStyler.set_table_styles([dict(selector='th', props=[('text-align', 'left')])])
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Convert Categorical variable to Numerical Variable¶
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for feature in vehicle.columns:# Loop through all columns in the dataframe
if vehicle[feature].dtype == 'object': # Only apply for columns with categorical strings
vehicle[feature] = pd.Categorical(vehicle[feature]) #Replace strings with an integer value
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replaceStruct = {"class": {"bus": 1,"car": 2, "van": 3}}
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# Update Original data values with categorical values
vehicle_new = vehicle.replace(replaceStruct)
# Illustrate Top 10 rows of modified data
vehicle_new.head(10)
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Correlation analysis: Pairplots and Correlation (with heatmap)¶
PairPlot: To plot multiple pairwise bivariate distributions in a dataset, the pairplot() function is used. This creates a matrix of axes and shows the relationship for each pair of columns in a DataFrame. By default, it also draws the univariate distribution of each variable on the diagonal Axes.
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sns.pairplot(vehicle_new, hue="class", diag_kind = 'kde', palette="husl", markers=["s", "*", "+"])
plt.show()
# Class Legend: s (value = 1) - bus, * (value = 2) - car, + (value = 3) - van
Pair Panel Analysis¶
- High degree of correlation exists amongst vehicle attributes such as compactness, circularity, radius_ratio, scatter_ratio, elongatedness, pr.axis_rectangularity, scaled_variance, scaled_variance.1, scaled_radius_of_gyration
- To measure degree of correlation, correlation analysis using pearson coefficient is performed
Correlation Coefficient¶
- The correlation coefficient is also known as the Pearson product-moment correlation coefficient, is obtained by dividing the covariance of the two variables by the product of their standard deviations.
- The values of the correlation coefficient can range from -1 to +1. The closer it is to +1 or -1, the more closely are the two variables are related. The positive sign signifies the direction of the correlation i.e. if one of the variables increases, the other variable is also supposed to increase.
- Now, if we look at the individual elements of the correlation matrix, the main diagonal all comprises of 1. This indicates that the correlation of an element with itself is 1, or the highest value possible.
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# Correlation Matrix Table
corr = vehicle_new.corr()
cmap = cmap=sns.diverging_palette(255, 5, as_cmap=True)
def magnify():
return [dict(selector="th",
props=[("font-size", "9pt")]),
dict(selector="td",
props=[('padding', "0em 0em")]),
dict(selector="th:hover",
props=[("font-size", "12pt")]),
dict(selector="tr:hover td:hover",
props=[('max-width', '100px'),
('font-size', '12pt')])
]
corr.style.background_gradient(cmap, axis=1)\
.set_properties(**{'max-width': '100px', 'font-size': '9pt'})\
.set_caption("Hover to Magify")\
.set_precision(2)\
.set_table_styles(magnify())
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# Representing coorelation through heatmap
fig, ax = plt.subplots(figsize=(20,12))
sns.heatmap(corr, center=0, cmap=cmap, annot=True, annot_kws={"size": 13})
plt.show()
Correlation Inferences
- High degree of correlation exists across variables except pr.axis_aspect_ratio, scaled_ratio_of_gyration.1, skewness_about, skewness_about.1, skewness_about.2, hollows_ratio, and class (i.e. Corr. Coeff.: > 0.70)
- Correlation of circularity to max.length_rectangularity and scaled_radius_of_gyration variables, Correlation of distance_circularity to scatter_ratio, elongatedness, pr.axis_rectangularity and scaled_variance.1 variables, Correlation of scatter_ratio to distance_circularity, pr.axis_rectangularity, scaled_variance, and scaled_variance.1 variables, Correlation of pr.axis_rectangularity to scaled_varaince and scaled_variance.1 variables are very very high (i.e. Corr. Coeff. >0.90)
- Elongatedness is highly negatively correlated with most data attributes viz. compactness, circularity, distance_circularity, radius_ratio, scatter_ratio, pr.axis_rectangularity, max.length_rectangularity, scaled_variance, scaled_variance.1, scaled_radius_of_gyration (i.e. Corr. Coeff.: >-0.75)
As variables are highly correlated or co-linear and hence can cause model overfitting or might run into the multicollinearity conundrum. Also, in plain English if two variables are so highly correlated they will obviously impart nearly exactly the same information to the model and by including both model will be weakened. No incremental information is added to the model instead model is infused with noise. Thus, pair of variables which are highly correlated can be removed to reduce dimensionality without much loss of information.
Dropping correlated variables from vehicle data:
- Threshold considered - Correlation >= 0.90 (Correlation equal to or greater than 90%)
- circularity has high degree of correlation with max.length_rectangularity (0.97), and scaled_radius_of_gyration (0.93), hence these variables are dropped
- distance_circularity has high degree of correlation with scatter_ratio (0.91), elongatedness (-0.91), pr.axis_rectangularity (0.90), and scaled_variance.1 (0.90), hence these variables are dropped
- scaled_radiuus_of_gyration.1 has high degree of correlation with hollows_ratio (-0.92), hence hollows_ratio variable is dropped
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vehicle_fn = vehicle_new.drop(['max.length_rectangularity', 'scaled_radius_of_gyration', 'scatter_ratio',
'elongatedness','pr.axis_rectangularity','scaled_variance.1', 'hollows_ratio'],
axis = 1)
vehicle_fn.shape
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corr1 = vehicle_fn.corr()
cmap = cmap=sns.diverging_palette(255, 5, as_cmap=True)
fig, ax = plt.subplots(figsize=(14,8))
sns.heatmap(corr1, center=0, cmap=cmap, annot=True, annot_kws={"size": 13})
plt.show()
Depending upon data model requirements, threshold cut-off for correlated variables can be selected to remove multi-collinearity in the data.
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